Non-Hausdorff Topology and Domain Theory

Selected Topics in Point-Set Topology

Jean Goubault-Larrecq

Table of Contents

1 Introduction page 6

2 Elements of Set Theory 9

2.1 Foundations 9

2.2 Finiteness, Countability 11

2.3 Order Theory 13

2.4 The Axiom of Choice 20

3 A First Tour of Topology: Metric Spaces 22

3.1 Metric Spaces 22

3.2 Convergence, Limits 24

3.3 Compact Subsets 30

3.4 Complete Metric Spaces 37

3.5 Continuous Functions 40

4 Topology 50

4.1 Topology, Topological Spaces 50

4.2 Order and Topology 58

4.3 Continuity 67

4.4 Compactness 71

4.5 Products 78

4.6 Coproducts 84

4.7 Convergence and Limits 86

4.8 Local Compactness 95

4.9 Subspaces 99

4.10 Homeomorphisms, Embeddings, Quotients, Retracts 101

4.11 Connectedness 110

4.12 A Bit of Category Theory I 114

5 Approximation, and Function Spaces 124

5.1 The Way-Below Relation 124

5.2 The Lattice of Open Subsets of a Space 144

5.3 Spaces of Continuous Maps 152

5.4 The Exponential Topology 155

5.5 A Bit of Category Theory II 172

5.6 C-Generated Spaces 183

5.7 Bc-Domains 197

6 Metrics, Quasi-Metrics, Hemi-Metrics 206

6.1 Metrics, Hemi-Metrics, and Open Balls 206

6.2 Continuous and Lipschitz Maps 211

6.3 Topological Equivalence, Hemi-Metrizability, Metrizability 217

6.4 Coproducts, Quotients 239

6.5 Products, Subspaces 241

6.6 Function Spaces 244

6.7 Compactness and Symcompactness 253

7 Completeness 261

7.1 Limits, d-Limits, and Cauchy Nets 261

7.2 A Strong Form of Completeness: Smyth-Completeness 268

7.3 Formal Balls 280

7.4 A Weak Form of Completeness: Yoneda-Completeness 289

7.5 The Formal Ball Completion 311

7.6 Choquet-Completeness 323

7.7 Polish Spaces 333

8 Sober Spaces 340

8.1 Frames and Stone Duality 340

8.2 Sober Spaces and Sobrification 350

8.3 The Hofmann-Mislove Theorem 363

8.4 Colimits and Limits of Sober Spaces 380

9 Stably Compact Spaces, and Compact Pospaces 396

9.1 Stably Locally Compact Spaces, Stably Compact Spaces 396

9.2 Coproducts and Retracts of Stably Compact Spaces 410

9.3 Products and Subspaces of Stably Compact Spaces 411

9.4 Patch-Continuous, Perfect Maps 417

9.5 Spectral Spaces 412

9.6 Bifinite Domains 438

9.7 Noetherian Spaces 451

References 472

List of illustrations 477

Index 479

Notation Index 487